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Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2008 Bosnia and Herzegovina Junior BMO TST
2008 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
3
1
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Nice geometry - paralellogram
Point
M
M
M
is given in the interior of parallelogram
A
B
C
D
ABCD
A
BC
D
, and the point
N
N
N
inside triangle
A
M
D
AMD
A
M
D
is chosen so that < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0. Prove that
M
N
MN
MN
is parallel to
A
B
AB
A
B
.
4
1
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Interesting circle result
On circle are
2008
2008
2008
blue and
1
1
1
red point(s) given. Are there more polygons which have a red point or those which dont have it??
1
1
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Interesting inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real positive numbers such that absolute difference between any two of them is less than
2
2
2
. Prove that: a \plus{} b \plus{} c < \sqrt {ab \plus{} 1} \plus{} \sqrt {ac \plus{} 1} \plus{} \sqrt {bc \plus{} 1}
2
1
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easy divisibility including 7
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be positive integers. If
7
7
7
divides (x\plus{}6y)(2x\plus{}5y)(3x\plus{}4y) than prove that
343
343
343
also divides it.